As is well known, if the multivariate function $f(\mathbf{x})$ is homogenous of degree $h$, then the partial derivatives of $f$ are homogenous of degree $h-1$. Also, say that we know $f$ is continuous. By Clairaut's theorem, then, the order in which differentiation is performed doesn't matter. This means that $f$ can be expressed as a quadratic form using its Hessian (denoted $\mathbf{M}$) as follows:
$$f=\frac{1}{h(h-1)}\mathbf{x'Mx}$$
But note that the second derivatives in $\mathbf{M}$ are homogenous of degree $h-2$. So I could go one step further and write:
$$f=\frac{1}{h(h-1)(h-2)}\mathbf{M}_{ijk}\mathbf{x}_i\mathbf{x}_j\mathbf{x}_k$$
Where I have (attempted) to resort to tensor notation and Einstein summation in order to extend the quadratic form and Hessian to third order.
My question is then how to generalize this to arbitrary higher orders. If $h$ is an integer, then $f$ is differentiable $h+1$ times, but if $h$ is non-integer, then $f$ is infinitely differentiable. I want a general expression for this case. In a recent post, I learned that the product $h(h-1)(h-2)...(h-n)$ is a "falling factorial" with its own notational convention. But I still need to clarify how the recursive matrix operation should be expressed. I would guess that it would be something like this (not worrying too much about how best to express the product for the moment):
$$f=\Pi_{i=0}^{n}\frac{1}{(h-i)}\mathbf{M}_{1,2,...,n}\mathbf{x}_1\mathbf{x}_2...\mathbf{x}_n ; n\rightarrow \infty$$
But that looks pretty awkward and a little ambiguous. Please educate me on how best to express this.